What lighting and background will make a realistic gemstone?
Lighting gems is an art that requires lots of patience, and what works for some scenes might not work for others.

With glass objects you are dealing mostly with objects that do not require to be lit, but do require other objects to be reflected on their surfaces, or seen trough them. In other words glass and crystals main attributes are that their surfaces are reflective, transparent and refractive.

So you need to create an environment that is conducive to show all of this characteristics.

Choose a background that will make your object stand out.

Create nice reflections:

Make an environment that is not homogeneous (like a flat color), but that has variations, for example a voronoi texture like in this setup: 





Or create a World that has some texture but that is not necessarily visible to the camera.





To show the transparency, place an object behind the glass (you can choose to make it invisible to the camera by turning off ray visibility)

 

To add more accents add other emissive plane objects with different sizes and shapes and intensities, and place them so that they can be reflected on the faces of the crystal at certain angles.



Rotating the object will make those objects reflect differently:



After that you can add post-production effects on the compositor, like glares, streaks etc.

All of this examples were done using the default glass shader and no lights.